24:null 25:["$","title",null,{"children":"Planets: Reconstructing 16th Century Orbits Using Tycho Brahe's Observations | Christophe Yamahata | Observable"}] 2a:T4ab,

This notebook is a continuation of a previous notebook which primarily focused on the orbit of Mars for the period 1582–1600.

Here, we showcase results for various planets by juxtaposing Tycho Brahe's meticulous observations of these planets with their respective estimated osculating orbital elements. Through this process, we aim to reconstruct the orbits of these planets as they unfolded at the close of the 16^th century.

Select a planet from the list to explore and compare its estimated orbit with the observational data recorded by Tycho Brahe. You can also adjust the number of iterations, currently set to ${iterations}, to observe its impact on the accuracy of the fitting curve.

2b:T8c1,{ let dotplot = Plot.plot({ height: 700, width: 1200, grid: true, marginLeft: 50, marginRight: 30, marginTop: 50, marginBottom: 80, insetTop: 0, insetBottom: 0, style: { fontSize: "13px", color: "#000" }, x: { label: " Year → ", tickPadding: 10, labelOffset: 70, tickFormat: "f", tickSize: 9, ticks: 10, domain: [Math.round(Math.min(...best_fit_curve.map(d => d.Date)),0), Math.round(Math.max(...best_fit_curve.map(d => d.Date)),0)], tickRotate: -35 }, y: { label: "↑ Declination, δ", tickPadding: 6, labelOffset: 30, tickFormat: x => x<=0 ?`${x}°`: `+${x}°`, ticks: 10, tickSize: 6, domain: [-30, 30] }, caption: html`
Figure: Tycho Brahe's observations of the declination of ${planet} (1582 – 1600) vs. theory (period ${Math.round(Math.min(...best_fit_curve.map(d => d.Date)),0)} – ${Math.round(Math.max(...best_fit_curve.map(d => d.Date)),0)}).
⚪ Tycho Brahe's observation between 1582 and 1600.
– – – Declination of ${planet} derived from JPL DE440 orbital elements for year ${reference_date}.
—— Optimized fitting curve (least squares regression employed in this notebook).
`, marks: [ Plot.frame({stroke: "black"}), Plot.ruleY([0]), inputs_graph_display.show_initial_curve? Plot.line(initial_curve, {x: "Date", y: "Declination", curve: "catmull-rom", stroke: "red", strokeWidth: 0.8, strokeOpacity: 0.8, strokeDasharray: "4,4"}):null, inputs_graph_display.show_best_fit_curve? Plot.line(best_fit_curve, {x: "Date", y: "Declination", curve: "catmull-rom", stroke: "darkblue", strokeWidth: 0.8, strokeOpacity: 0.8}):null, inputs_graph_display.show_data_points? Plot.dot(data_Tycho_Brahe, {x: "Date", y: "Declination", stroke:"#111", fill: "orange", strokeOpacity: 0.4, r: dot_size}):null, ] }) // Label in bold/large font let select_labels = d3.select(dotplot) .selectAll("svg > g > text") .style("font-weight", "bold") .style("font-size", "16px"); return dotplot; }2c:T82c, ## Useful links These Observable notebooks are dedicated to the following observations: - [Tycho Brahe's observations of Mars (♂)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahe-s-astronomical-observations-mars?collection=@christophe-yamahata/tycho-brahe-and-mars) ★★★ → In this notebook and the associated collection, you can visualize Tycho Brahe's astronomical observations of Mars. - [Tycho Brahe's observations of Mercury (☿)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-mercury?collection=@christophe-yamahata/astronomy) → A fork of the previous notebook, featuring data collected by Brahe for the planet Mercury. - [Tycho Brahe's observations of Venus (♀)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-venus?collection=@christophe-yamahata/astronomy) → This notebook features the data collected by Brahe for the planet Venus. - [Tycho Brahe's observations of Jupiter (♃)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-jupiter?collection=@christophe-yamahata/astronomy) → This notebook features the data collected by Brahe for the planet Jupiter. - [Tycho Brahe's observations of Saturn (♄)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-saturn?collection=@christophe-yamahata/astronomy) → This notebook features the data collected by Brahe for the planet Saturn. - [Reconstructing 16^th Century Planetary Orbits Using Tycho Brahe's Observations and Joint Optimization](https://observablehq.com/@christophe-yamahata/reconstructing-16th-century-planetary-orbits-using-tycho-brahe-observations-and-joint-optimization?collection=@christophe-yamahata/astronomy) ★★★ → In this notebook, we utilize joint optimization to derive the optimal osculating orbital elements for all the planets.2d:T71e,

☿ – Tycho Brahe's observations of the planet Mercury (period 1585–1600)

♀ – Tycho Brahe's observations of the planet Venus (period 1582–1600)

♂ – Tycho Brahe's observations of the planet Mars (period 1582–1600)

♃ – Tycho Brahe's observations of the planet Jupiter (period 1582–1600)

♄ – Tycho Brahe's observations of the planet Saturn (period 1582–1600)2e:T6e2,// Model to compute the declination of planet Mars. model = function(x, p){return x.map(function(x_i){ const [a_earth, e_earth, i_earth, n_earth, o_earth, w_earth, M_earth, a_planet, e_planet, i_planet, n_planet, o_planet, w_planet, M_planet, obliquity_ecliptic] = p; const obliquity_rad = obliquity_ecliptic*Math.PI/180; const p_earth = o_earth + w_earth; const L_earth = p_earth + M_earth; const p_planet = o_planet + w_planet; const L_planet = p_planet + M_planet; const D = (x_i - reference_date) * 365.25; const MA_earth = n_earth * D + L_earth - p_earth; const MA_planet = n_planet * D + L_planet - p_planet; const v_earth = true_anomaly(MA_earth, e_earth); const v_earth_rad = v_earth*Math.PI/180; const v_planet = true_anomaly(MA_planet, e_planet); const v_planet_rad = v_planet*Math.PI/180; const r_earth = a_earth * (1 - Math.pow(e_earth,2)) / (1 + e_earth * Math.cos(v_earth_rad)); const r_planet = a_planet * (1 - Math.pow(e_planet,2)) / (1 + e_planet * Math.cos(v_planet_rad)); const { X: X_earth, Y: Y_earth, Z: Z_earth } = heliocentric_coordinates(r_earth, v_earth, o_earth, p_earth, i_earth); const { X: X_planet, Y: Y_planet, Z: Z_planet} = heliocentric_coordinates(r_planet, v_planet, o_planet, p_planet, i_planet); const X_prime = X_planet - X_earth; const Y_prime = Y_planet - Y_earth; const Z_prime = Z_planet - Z_earth; const X_q = X_prime; const Y_q = Y_prime * Math.cos(obliquity_rad) - Z_prime * Math.sin(obliquity_rad); const Z_q = Y_prime * Math.sin(obliquity_rad) + Z_prime * Math.cos(obliquity_rad); const DEC = Math.atan(Z_q/Math.sqrt(Math.pow(X_q,2)+Math.pow(Y_q,2)))*180/Math.PI; return DEC; })}2f:T948,// Modified fminsearch function with bounds: Please note that this implementation is not optimized. // Adapted from: https://github.com/jonasalmeida/fminsearch // See associated notebook: https://observablehq.com/@christophe-yamahata/nonlinear-regression-with-bounds-in-javascript fminsearch = function(fun, Parm0, x, y, Opt) { if (!Opt) { Opt = {} }; if (!Opt.maxIter) { Opt.maxIter = 1000 }; if (!Opt.step) { // initial step is 1/100 of initial value (remember not to use zero in Parm0) Opt.step = Parm0.map(function(p) { return p / 100 }); Opt.step = Opt.step.map(function(si) { if (si == 0) { return 1 } else { return si } }); // convert null steps into 1's }; if (typeof (Opt.display) == 'undefined') { Opt.display = true }; if (!Opt.objFun) { Opt.objFun = function(y, yp) { return y.map(function(yi, i) { return Math.pow((yi - yp[i]), 2) }).reduce(function(a, b) { return a + b }) } } //SSD if (!Opt.bounds) { Opt.bounds = Parm0.map(function() { return [-Infinity, Infinity] }) }; // Default no bounds var cloneVector = function(V) { return V.map(function(v) { return v }) }; var ya, y0, yb, fP0, fP1; var P0 = cloneVector(Parm0), P1 = cloneVector(Parm0); var n = P0.length; let step = cloneVector(Opt.step); // Ensure step is reinitialized var funParm = function(P) { return Opt.objFun(y, fun(x, P)) } //function (of Parameters) to minimize // Apply bounds var applyBounds = function(P, bounds) { return P.map(function(p, i) { return Math.max(bounds[i][0], Math.min(bounds[i][1], p)); }); } // Silly multi-univariate screening for (var i = 0; i < Opt.maxIter; i++) { for (var j = 0; j < n; j++) { // take a step for each parameter P1 = cloneVector(P0); P1[j] += step[j]; P1 = applyBounds(P1, Opt.bounds); // Apply bounds after updating the parameter if (funParm(P1) < funParm(P0)) { // if parm value going in the right direction step[j] = 1.2 * step[j]; // then go a little faster P0 = cloneVector(P1); } else { step[j] = -(0.5 * step[j]); // otherwise reverse and go slower } } if (Opt.display) { if (i > (Opt.maxIter - 10)) { console.log(i + 1, funParm(P0), P0) } } } return P0 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Map(planets.map(planet => [planet.display, planet.name])), \n { format: x => x[0], value: \"Jupiter\", label: htl.html`Planet`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":256,"value":"viewof inputs_graph_display = columns({\n show_data_points : Inputs.toggle({label: \"Tycho Brahe's observations\", value: true}),\n show_initial_curve : Inputs.toggle({label: htl.html`Declination derived from JPL DE440`, value: false}),\n show_best_fit_curve : Inputs.toggle({label: \"Optimized fitting curve (least squares regression)\", value: true}), \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":547,"value":"display_dot_properties = {\n const html = inputs_graph_display.show_data_points? htl.html`${viewof dot_size}`: htl.html`

`; \n yield html\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":617,"value":"viewof iterations = Inputs.range([5, 200], {value: 50, step: 5, label: html`Num. of iterations
(least squares regression)

`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"$2b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1527,"value":"
\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1510,"value":"$2c","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1622,"value":"$2d","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"

Appendix

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":802,"value":"

Tycho Brahe's observations

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1673,"value":"// Tycho Brahe's observation of Mercury (Declination) for the period 1585-1600\nMercury_data_Tycho_Brahe = FileAttachment(\"Mercury_data_1585-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":772,"value":"// Tycho Brahe's observation of Venus (Declination) for the period 1582-1600\nVenus_data_Tycho_Brahe = FileAttachment(\"Venus_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"// Tycho Brahe's observation of Mars (Declination) for the period 1582-1600\nMars_data_Tycho_Brahe = FileAttachment(\"Mars_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":785,"value":"// Tycho Brahe's observation of Jupiter (Declination) for the period 1582-1600\nJupiter_data_Tycho_Brahe = FileAttachment(\"Jupiter_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":793,"value":"// Tycho Brahe's observation of Saturn (Declination) for the period 1582-1600\nSaturn_data_Tycho_Brahe = FileAttachment(\"Saturn_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1676,"value":"// data: x = dates as decimal years\nx_Mercury = {\n let data = [];\n for (let i = 0; i < Mercury_data_Tycho_Brahe.length; i++) {\n data.push(Mercury_data_Tycho_Brahe[i].Date);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1679,"value":"// data: y = declination of Mercury in degrees\ny_Mercury = {\n let data = [];\n for (let i = 0; i < Mercury_data_Tycho_Brahe.length; i++) {\n data.push(Mercury_data_Tycho_Brahe[i].Declination);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":776,"value":"// data: x = dates as decimal years\nx_Venus = {\n let data = [];\n for (let i = 0; i < Venus_data_Tycho_Brahe.length; i++) {\n data.push(Venus_data_Tycho_Brahe[i].Date);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":781,"value":"// data: y = declination of Venus in degrees\ny_Venus = {\n let data = [];\n for (let i = 0; i < Venus_data_Tycho_Brahe.length; i++) {\n data.push(Venus_data_Tycho_Brahe[i].Declination);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"// data: x = dates as decimal years\nx_Mars = {\n let data = [];\n for (let i = 0; i < Mars_data_Tycho_Brahe.length; i++) {\n data.push(Mars_data_Tycho_Brahe[i].Date);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"// data: y = declination of Mars in degrees\ny_Mars = {\n let data = [];\n for (let i = 0; i < Mars_data_Tycho_Brahe.length; i++) {\n data.push(Mars_data_Tycho_Brahe[i].Declination);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":787,"value":"// data: x = dates as decimal years\nx_Jupiter = {\n let data = [];\n for (let i = 0; i < Jupiter_data_Tycho_Brahe.length; i++) {\n data.push(Jupiter_data_Tycho_Brahe[i].Date);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":790,"value":"// data: y = declination of Jupiter in degrees\ny_Jupiter = {\n let data = [];\n for (let i = 0; i < Jupiter_data_Tycho_Brahe.length; i++) {\n data.push(Jupiter_data_Tycho_Brahe[i].Declination);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":796,"value":"// data: x = dates as decimal years\nx_Saturn = {\n let data = [];\n for (let i = 0; i < Saturn_data_Tycho_Brahe.length; i++) {\n data.push(Saturn_data_Tycho_Brahe[i].Date);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":799,"value":"// data: y = declination of Saturn in degrees\ny_Saturn = {\n let data = [];\n for (let i = 0; i < Saturn_data_Tycho_Brahe.length; i++) {\n data.push(Saturn_data_Tycho_Brahe[i].Declination);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":989,"value":"// Combined data for Mercury, Venus, Mars, Jupiter, and Saturn\nplanets_data = [\n { name: 'Mercury', x: x_Mercury, y: y_Mercury },\n { name: 'Venus', x: x_Venus, y: y_Venus },\n { name: 'Mars', x: x_Mars, y: y_Mars },\n { name: 'Jupiter', x: x_Jupiter, y: y_Jupiter },\n { name: 'Saturn', x: x_Saturn, y: y_Saturn }\n];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1393,"value":"data_Tycho_Brahe = createDataForPlanet(planets_data, planet)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":965,"value":"

Other parameters

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":76,"value":"// The initial values for the osculating orbital elements (stored in `p_init`)\n// were obtained for the reference date: January 1, 1600 (Julian Date, JD = 2305447.5)\nreference_date = 1600","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"// Maximum iterations for the nonlinear least squares optimization\nmax_iterations = iterations","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1535,"value":"planets = [\n {name: \"Mercury\", display: \"Mercury (☿)\"}, \n {name: \"Venus\", display: \"Venus (♀)\"}, \n {name: \"Mars\", display: \"Mars (♂)\"},\n {name: \"Jupiter\", display: \"Jupiter (♃)\"},\n {name: \"Saturn\", display: \"Saturn (♄)\"},\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1641,"value":"image_width = 640","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":565,"value":"viewof dot_size = Inputs.range([0.4, 4], {value: 2, step: 0.2, label: \"Dot size:\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1017,"value":"// Initial guess for obliquity of the ecliptic\nobliquity_ecliptic = 23.491553","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1005,"value":"// Initial guess for Earth's parameters\np_init_Earth = [1.000001, 0.016853, 0.052404, 0.985607, 355.777916, 105.937340, 3.725488]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1684,"value":"// Initial guess for Mercury's parameters\np_init_Mercury = [0.38710, 0.20554, 7.02873, 4.09233, 48.83104, 27.99055, 254.96184]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1010,"value":"// Initial guess for Venus's parameters\np_init_Venus = [0.723344, 0.006963, 3.397583, 1.602092, 77.790662, 53.655314, 343.275912]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1002,"value":"// Initial guess for Mars's parameters\np_init_Mars = [1.523777, 0.092999, 1.88184, 0.523989, 50.727771, 283.539037, 141.282667]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1021,"value":"// Initial guess for Jupiter's parameters \np_init_Jupiter = [5.201876,0.048003,1.311245,0.083113,99.741520,272.718863,122.623172]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1025,"value":"// Initial guess for Saturn's parameters\np_init_Saturn = [9.572787, 0.057747, 2.479350, 0.033282, 114.683987, 340.112017, 106.440683]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1164,"value":"// Combined initial parameters for Earth, Mercury, Venus, Mars, Jupiter, and Saturn\np_init_list = [\n { name: 'Earth', p_init: p_init_Earth },\n { name: 'Mercury', p_init: p_init_Mercury },\n { name: 'Venus', p_init: p_init_Venus },\n { name: 'Mars', p_init: p_init_Mars },\n { name: 'Jupiter', p_init: p_init_Jupiter },\n { name: 'Saturn', p_init: p_init_Saturn }\n];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1472,"value":"initial_curve = create_fitting_curve(get_p_init(planet))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":199,"value":"best_fit_curve = create_fitting_curve(fminsearchWrapper(planet, model, get_p_init(planet), Opt(planet)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":963,"value":"

JavaScript functions

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"// Calculate true anomaly from mean anomaly and eccentricity.\n// tex`v = M + \\frac{180}{\\pi} \\times [ (2e - \\frac{1}{4}e^3) \\times \\sin(M) \n// + \\frac{5}{4}e^2 \\times \\sin(2M) + \\frac{13}{12} e^3 \\times \\sin(3M) ]`\ntrue_anomaly = function(M, e){\n var M_rad = M*Math.PI/180;\n return M + (180 / Math.PI) * ((2 * e - Math.pow(e,3)/4) * Math.sin(M_rad)\n + (5 / 4) * Math.pow(e,2) * Math.sin(2 * M_rad)\n + (13 / 12) * Math.pow(e,3) * Math.sin(3 * M_rad))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":100,"value":"// Calculate heliocentric coordinates.\nheliocentric_coordinates = function(r, v, o, p, i){\n const v_rad = (v + p - o)*Math.PI/180;\n const o_rad = o*Math.PI/180;\n const i_rad = i*Math.PI/180;\n \n const X = r * (Math.cos(o_rad) * Math.cos(v_rad) - Math.sin(o_rad) * Math.sin(v_rad) * Math.cos(i_rad));\n const Y = r * (Math.sin(o_rad) * Math.cos(v_rad) + Math.cos(o_rad) * Math.sin(v_rad) * Math.cos(i_rad));\n const Z = r * Math.sin(v_rad) * Math.sin(i_rad);\n \n return {X, Y, Z};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":980,"value":"// Define bounds for parameters\nfunction bounds(planet) {\n const p_init = get_p_init(planet);\n return [ // [lower, upper] bounds\n /* Bounds for Earth's parameters */\t\n [0.998 * p_init[0], 1.002 * p_init[0]], \n [0.999 * p_init[1], 1.001 * p_init[1]], \n [0.999 * p_init[2], 1.001 * p_init[2]], \n [0.999 * p_init[3], 1.001 * p_init[3]], \n [0, 360], [0, 360], [0, 360],\n\n /* Bounds for planet's parameters */\n [0.998 * p_init[7], 1.002 * p_init[7]], \n [0.999 * p_init[8], 1.001 * p_init[8]], \n [0.999 * p_init[9], 1.001 * p_init[9]], \n [0.999 * p_init[10], 1.001 * p_init[10]], \n [0, 360], [0, 360], [0, 360],\n\n /* Bounds for obliquity of the ecliptic */\n [Math.min(23.4, obliquity_ecliptic), Math.max(23.5, obliquity_ecliptic)]\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"// Define the bounding parameters for nonlinear regression calculation using bounded parameters.\nfunction Opt(planet) {\n return {\n maxIter: max_iterations, // Optional: Specify the maximum number of iterations\n bounds: bounds(planet), // Bounds for the parameters\n }\n};","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":81,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1191,"value":"// Wrapper function to simplify model call\nfunction modelWrapper(planetName) {\n const x_data = getPlanetData(planets_data, 'x', planetName);\n const p_init = get_p_init(planetName);\n return model(x_data, p_init);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1118,"value":"function getPlanetData(planets_data, d, planetName) {\n const planetEntry = planets_data.find(entry => entry.name === planetName);\n return planetEntry[d];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1167,"value":"// Return an array of the form [p_init_Earth, p_init_Planet, obliquity_ecliptic]\nfunction get_p_init(planetName) {\n const p_init_Earth = p_init_list.find(entry => entry.name === 'Earth').p_init;\n const p_init_Planet = p_init_list.find(entry => entry.name === planetName).p_init;\n return p_init_Earth.concat(p_init_Planet, obliquity_ecliptic);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1384,"value":"function createDataForPlanet(planets_data, planetName) {\n const planetEntry = planets_data.find(entry => entry.name === planetName);\n if (planetEntry) {\n const data = planetEntry.x.map((date, index) => ({\n Date: date,\n Declination: planetEntry.y[index]\n }));\n return data;\n } else {\n throw new Error(`No data found for planet: ${planetName}`);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":195,"value":"function create_fitting_curve(parameters){\n const data = [];\n const n = 1500;\n const year_min = 1574;\n const year_max = 1602;\n\n const years = d3.range(year_min, year_max, (year_max - year_min) / n).map(d => d.toFixed(4));\n const DEC = model(years,parameters);\n\n for (let i = 0; i < years.length; i++) {\n data.push({\n Date: parseFloat(years[i]),\n Declination: DEC[i]\n });\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":709,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1200,"value":"// Wrapper function to simplify fminsearch call for a specific planet\nfunction fminsearchWrapper(planetName, model, p_init, Opt) {\n const x_data = getPlanetData(planets_data, 'x', planetName);\n const y_data = getPlanetData(planets_data, 'y', planetName);\n const p_init_combined = get_p_init(planetName);\n return fminsearch(model, p_init_combined, x_data, y_data, Opt);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":258,"value":"// https://observablehq.com/@bcardiff/observable-columns\ncolumns = (args) => {\n const form = html``\n form.value = {}\n\n let cols = 0\n for (const key in args) {\n form.appendChild(args[key])\n cols++\n }\n \n form.style = `width:200px; display: grid; grid-gap: 10px 100px; grid-template-columns: repeat(${cols}, auto); grid-auto-flow: row;`\n \n form.oninput = () => {\n form.value = Object.keys(args).reduce((result, key) => {\n result[key] = args[key].value\n return result\n }, Array.isArray(args) ? [] : {})\n }\n form.oninput()\n \n return form\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W30","stdlib":"1","titling":"auto"}} 29:"$28:body"